Recently σ-models have received a lot of attention for many reasons. One interesting aspect of the CP(^n) sigma models is the fact they are the simplest Lorentz invariant models which possess topologically stable (minimum of the action) solutions in (2+0) dimensions. Unfortunately, it appears that Lorentz covariance and integrability are incompatable in (2+1) dimensions. In the literature a few integrable models were constructed in (2+1) dimensions at the expense of Lorentz invariance (e.g. modified chiral model,...). An alternative way to proceed is to retain Lorentz invariance and relax the property of integrability by replacing it with a new property of quasi-integrability. Zakrzewski and others have constructed an example of such quasi-integrable models. Their example is based on the CP(^1) model modified by the addition of two stabilising terms (the first called the "Skyrme-like" term and the second the "potential-like" term) to the basic Lagrangian. In this thesis we have addressed the following relevant questions: How unique is this model? What are the properties of its static structures (skyrmions)? Is it possible to generalise this model? Is quasi-integrabilty, as a property, shared by all CP(^2) models, or it is only restricted to the CP(^1) model? It turns out that the first stabilising term [i.e the Skyrme-like term) is only unique for CP(^1) model and this uniqueness does not survive the generalisations to larger coset spaces, say, CP(^n). The second stabilising term is not unique. By taking advantage of this observation, i.e arbitrariness of the potential term, a generalisation of Zakrzewski's model has become possible. Most important of all is the fact that all the CP" models are quasi-integrable provided one incurs the size instabilities of their soliton solutions.

In this work we apply the field theoretical Schrodinger representation to the massive ϕ⁴ theory and the O(N) σ model in 1 + 1 dimensions. The Schrodinger equation for the ϕ⁴ theory is reviewed and then solved classically and semiclassically, to obtain the vacuum functional as an expansion of local functionals. These results are compared with equivalent ones derived from the path integral formulation to prove their agreement with the conventional field theoretical methods. For the O(N) a model we construct the functional Laplacian, which is the principal ingredient of the corresponding Schrodinger equation. This result is used to construct the generalised Virasoro operators for this model and study their algebra.

This thesis is mainly concerned with monopoles. First, the existence of monopoles and their behaviour in the Yang-Mills-Higgs theories, and in parallel, the instanton solutions of the Yang-Mills fields are explained. One part of this work is about monopoles and instantons in low-energy string theory. A general instanton solution for the heterotic string theory is obtained by using the ADHM construction for the classical subgroups of the string gauge group. In this direction, the embedding of subgroups and a general formula for the dilaton are explained. In the next topic of this part, the i/-monopole and its generalisation to different subgroups of the string gauge group are discussed. In the second part, the motion of the Yang-Mills particles in the Yang-Mills- Higgs fields are studied. Planar orbits are observed for a particle in a monopole field when the Higgs field contribution is neglected. The planar orbits are studied further with some numerical analysis of the equations of motion. By regarding the Higgs field contribution, a complete set of equations are worked out for the particle and fields. In this scenario, the planar motions as well as three-dimensional bounded motions are studied. At the end, the force exerted by the non-abelian Yang-Mills-Higgs fields on a particle with non-abelian charge is explored.

In 1986 Polyakov published his theory of rigid string. I investigate the instantons associated with the consequent new fine structure of strings in four dimensional Euclidean space-time. I reduce the self-dual equation of rigid string instantons to a simple form and show that (p,q) torus knots satisfy the equation, thus forming an interesting new class of solutions. I calculate by computer the world-sheet self-intersection number of the first few such closed knotted strings and derive a very simple formula for the self-intersection number of a torus knot. I consider an interpretation in terms of the first Chem number and discover the empirical formula Q = q - p for the inslanton number, Q, of torus knots and links. In 1987 Biran, Floratos and Savvidy pioneered an approach for constructing self-dual equations for membranes. I present some new solutions for self-dual membranes in three dimensions. In 1989 Grabowski and Tze pointed out a new class of exceptional immersions for which self-dual equations can be constructed and for which there are no known non-trivial solutions. By analogy with (p,q) torus knots, I describe an algorithm for generating a class of potential solutions of self-dual lumps in eight dimensions. I show how these come to within a single sign change of solving all the required constraints and come very close to solving all the 32 self-dual (4;8)-brane equations.

This work is concerned with the breaking of chiral symmetry in gauge theories and the associated generation of a dynamical mass scale. We investigate this phenomenon in the context of a simple model, three dimensional QED, where the complicating factor of infinite renormalisations is absent. This model possesses an intrinsic scale, set by the coupling [e(^2)] = M, and it is the relationship between this and the dynamically generated mass scale that is of interest. The chiral symmetry breaking mechanism is investigated using the Schwinger Dyson equations which are then truncated in a nonperturbative manner using the Ball-Chiu vertex ansatz. The complexity of the resulting coupled fermion-photon system means that the photon is initially replaced by its perturbative form. Numerical investigations of this simplified system then reveal the existence of an exponential relationship, in terms of the dimensionless parameter N, between the intrinsic and dynamical mass scales, m ~ e(^2) exp(-cN). Contrary to the assertions of Appelquist et al the wavefunction renormalisation was found to be nonperturbative and crucial in determining this behaviour. The sensitivity of this mechanism to the nonperturbative behaviour of the photon is investigated. A simple analysis shows it to be far stronger than previously expected. This is confirmed by a numerical analysis of the coupled photon-fermion system which suggest the relationship between the two scales in the theory is of the form m ~ e(^2) exp(-cN(^2)). This model therefore illustrates how a large hierarchy of scales may naturally occur in a gauge theory, for instance N=3 m/a ~ 10(^-5). Finally an investigation of the gauge dependence of this approach is initiated. The softening of the photon in the low momentum region is shown to amplify automatically any inadequacy of the vertex ansatz by factors of O(a/m) in all but the Landau gauge. It is therefore expected that any incomplete vertex form will result in the generation of a "critical gauge", ɛ(_e), below which chiral symmetry breaking solutions will not exist. A path of further investigation is suggested.

The rôle that shear plays in the dynamical response and associated stability of elastic bodies is investigated within this thesis from two perspectives. Forming the major part of the study is the investigation of infinitesimal wave propagation within elastic material which has been subjected to a static pre-strain corresponding to simple shear. Initially we consider a prototype problem wherein the theory of incremental motions provides the mechanism for analyzing Rayleigh waves propagating along the surface of an incompressible elastic half-space. This is looked at from a plane strain point of view but, significantly, the direction of propagation is not along a principal axis. Using co-ordinates measured relative to the Eulerian axes in the governing equation and boundary conditions, corresponding to the vanishing of incremental tractions, we derived the secular equation for infinitesimal waves in terms of wavespeed, shear and hydrostatic stress parameters for a particular class of materials. The dependence of the wavespeed on these parameters is illustrated and bifurcation criteria are found through setting the wavespeed to zero, this corresponding to quasi-static incremental displacements. For a general form of incompressible, isotropic strain-energy function we are also able to provide the bifurcation criteria incorporating an additional, material parameter. We also consider the compressible counterpart to this problem and follow the same approach, where possible, in establishing the secular equation for compressible materials. This approach is also adopted for the next problem in which an infinite layer of incompressible elastic material, having uniform width, is pre-strained and within which infinitesimal waves are propagated along the layer. Owing to the layer width the waves are now dispersive and for three types of incremental boundary conditions we provide dispersion equations involving wavespeed, shear, hydrostatic stress and layer thickness (wavelength) parameters for the same particular class of materials. The interdependence of these parameters is comprehensively detailed for this class along with the bifurcation analysis which is again extended so that it may be applicable to a general incompressible, isotropic material.

Chapter 1 covers the background necessary for what follows. In particular, general properties of injectivity and some of its well-known generalizations are stated. Chapter 2 is concerned with two generalizations of injectivity, namely near and essential injectivity. These concepts, together with the notion of the exchange property, prove to be a key tool in obtaining characterizations of when the direct sum of extending modules is extending. We find sufficient conditions for a direct sum of two extending modules to be extending, generalizing several known results. We characterize when the direct sum of an extending module and an injective module is extending and when the direct sum of an extending module with the finite exchange property and a semisimple module is extending. We also characterize when the direct sum of a uniform-extending module and a semisimple module is uniform-extending and, in consequence, we prove that, for a right Noetherian ring R, an extending right R-module M1 and a semisimple right R-module M2, the right R-module M1 M2 is extending if and only if M2 is M1/Soc(M1)-injective. Chapter 3 deals with the class of self-c-injective modules, that can be characterised by the lifting of homomorphisms from closed submodules to the module itself. We prove general properties of self-c-injective modules and find sufficient conditions for a direct sum of two self-c-injective to be self-c-injective. We also look at self-cu-injective modules, i.e. modules M such that every homomorphism from a closed uniform submodule to M can be lifted to M itself. We prove that every self-c-injective free module over a commutative domain that is not a field is finitely generated and then proceed to consider torsion-free modules over commutative domains, as was done for extending modules in [31]. We also characterize when, over a principal ideal domain, the direct sum of a torsion-free injective module and a cyclic torsion module is self-cu-injective.

A number of existing methods, suitable for the analysis of periodic structures are reviewed. Most of these methods are related to the well known Transfer Matrix Method (TMM). It is seen that TMM, though efficient, suffers from a number of numerical stability problems which will prevent a solution being obtained in certain circumstances. Two lines of approach are selected for further development which make both numerical stability and solution efficiency possible. The first is an algebraic solution of the governing difference equations known as the Matrix Difference Method (MDM). Existing closed form solutions by MDM are given for box girders and for beams on elastic supports with axial and transverse loading. The latter is used for a parametric study and in the analysis of U-frame bridges. MDM is also used to investigate the characteristics of vierendeel girders, tall building frames and a counterbraced truss. An upper limit of problem size for MDM analysis is suggested. The second approach to periodic structure analysis is a numerical one in which the structure is modelled using finite elements. A new general method for the analysis of infinitely long periodic structures, the Deflection Transfer Matrix Method (DTMM), is developed. The method involves an iterative procedure, similar to Gaussian elimination, to find the reduced structure stiffness matrix. DTMM is programmed in FORTRAN and is interfaced with a comprehensive finite element program (LUSAS). Results for the analysis of a closed cylindrical shell and an infinite plate on elastic subgrade are compared with F.E. and published solutions. The method is appraised and suggestions for further developments are given in detail. Extending the numerical approach to finite length periodic structures, an eigenvalue method, used previously for the static analysis of deep beams, is developed. The theory is extended to make it general and the method is programmed in FORTRAN and interfaced with LUSAS. Results are obtained for deep beams and a large coupled shear wall, comparison being made with a normal F.E. solution and existing experimental and TMM solutions. The method (EST) is found to be versatile, stable and efficient. Suggestions for further development are made.

Let $V$ be a non-zero finite dimensional vector space over a finite field $\mathbb{F}_q$ of odd characteristic. Fixing a non-singular quadratic form $\xi_0$ in $S^2(V^*)$, the symmetric square of the dual of V we are concerned with the Orthogonal group $O(\xi_0)$, the subgroup of the General Linear Group $GL(V)$ that fixes $\xi_0$ and with invariants of this group. We have the Dickson Invariants which being invariants of the General Linear Group are then invariants of $O(\xi_0)$. Considering the $O(\xi_0)$ orbits of the dual vector space $\vs$ we generate the Chern Orbit polynomials, the coefficients of which, the Chern Orbit Classes, are also invariants of the Orthogonal group. The invariants $\xi_1, \xi_2, \dots $ are be generated from $\xi_0$ by applying the action of the Steenrod Algebra to $S^2(V^*)$ which being natural takes invariants to invariants. Our aim is to discover further invariants from these known invariants with the intention of establishing a set of generators for the the Ring of invariants of the Orthogonal Group. In particular we calculate invariants of $O(\xi_0)$ when the dimension of the vector space is $4$ the finite field is $\mathbb{F}_3$ and the quadratic form is $\xi_0=x_1^2+x_2^2+x_3^2+x_4^2$ and we are able to establish an explicit presentation of $O(\xi_0)$ in this case.

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